Social Choice with Fuzzy Preferences

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SOCIAL CHOICE WITH FUZZY PREFERENCES

RICHARD BARRETT AND MAURICE SALLES

1. introduction The most important concept of social choice theory is probably the concept of preference, be it individual preference or social preference. Typically, preferences are crisp, i.e., given by a binary relation over the set of options (social states, candidates, etc.). Then, for two options, either one is preferred to the other, or there is an indifference between them or there is no relation between them. In most cases, the possibility that there is no relation between them is excluded. The binary relation is then complete. Several authors, including Sen (1992), have considered that incompleteness was a way to deal with ambiguity. However, when the options are uncertain we must adopt preferences related in some way to probabilities and when they are complex, for instance when each option has manifold characteristics, we must try to take account of the vagueness this entails. To deal with vagueness, various authors have had recourse to, roughly speaking, three different techniques. The first is probability theory. Fishburn (1998) writes: ‘Vague preferences and wavering judgments about better, best, or merely satisfactory alternatives lead naturally to theories based on probabilistic preference and probabilistic choice.’ The second is fuzzy set theory. Each of these two theories has champions that apparently strongly disagree. There are, however, several signs indicating that the scientific war should eventually end (see Ross, Booker and Parkinson (2002) and the forewords in this book by Zadeh and by Suppes). The third is due to philosophers. Vagueness is an important topic within analytic philosophy (Keefe and Smith (1996), Burns (1991), Williamson (1994), Keefe (2000)). Philosophers have mainly been concerned with predicates such as tall, small, red, bald, heap, and solutions to the sorites paradox. (For instance, a nice example for the predicate ‘small’ is Wang’s paradox discussed by Dummett (1975). Consider the inductive argument: 0 is small; If n is small, n + 1 is small: Therefore every number is small.) Although there is little agreement among them about the nature of vagueness, a theory has been designed to deal with it: supervaluation theory. In an interesting paper, Broome (1997) has jointly considered vagueness and incompleteness.

As social choice theorists, we are perhaps more interested in vagueness of relations than the kind of predicates mentioned above and most of the papers dealing with vagueness in choice theory have used fuzzy sets. The purpose of this chapter is to present some of the main results obtained on the aggregation of fuzzy preferences. Regarding the possible interpretations of preferences, it seems to us that we must conﬁne ourselves to what Sen (1983, 2002) calls the outcome evaluation. Preference is about the fact that an option is judged to be a better state of aﬀairs than another option. In Sen’s typology, the two other interpretations are about choices, normative or descriptive. In particular, our analysis is not well adapted to voting, since the voters’ ballot papers are not vague and the results of the election are not vague either, even if, before designing a crisp preference, each voter had rather fuzzy preferences.

Date: August 10, 2004. We are most grateful to Prasanta Pattanaik for our joint work on this topic over the years. Thanks also to a referee for useful suggestions and to Dinko Dimitrov for his careful reading and for providing corrections. 1 2 RICHARD BARRETT AND MAURICE SALLES

After introducing the concepts of fuzzy preference in Section 2, Section 3 will be de- voted to Arrovian aggregation problems and Section 4 to other aspects, including a fuzzy treatment of Sen’s impossibility of a Paretian liberal and the ﬁrst results about fuzzy aggregation in economic environments.

2. fuzzy preferences In (naive) set theory, given a set X, a subset A and an element x ∈ X, either x ∈ A or x∈ / A. Belonging to the subset can be deﬁned by a function b from X to {0, 1}, where x ∈ A is equivalent to b(x) = 1 and x∈ / A is equivalent to b(x) = 0. If the subset A refers to the description of some semantic concepts (events or phenomena or statements), there might be no clear-cut way to assert that an element is or is not in this subset. Classical examples are the set of tall men, the set of intelligent women or the set of beautiful spiders. The basic idea of replacing {0, 1} by [0, 1] as the set where the membership function takes its values is due to Zadeh. However, the origin of this is probably older, taking us back at least toLukasiewicz who introduced many-valued logic in 1920. For excellent mathematical introductions, we recommend Dubois and Prade (1980) and Nguyen and Walker (2000).

For a fuzzy binary relation, the membership function associates a number in [0, 1] to an ordered pair of options (x, y). The interpretation of a number α ∈ [0, 1] associated to (x, y) can be the degree of intensity with which x is preferred to y is α or the degree of truth that x is preferred to y is α. Though in some cases the ﬁrst interpretation is possible and amounts to considering strength of preference, the second interpretation is always possible and is compulsory if, at some stage, fuzzy connectives (and, or ...) have been introduced.

The fact that some positive value α is associated to (x, y) does not entail, in general, that the value 0 must be assigned to (y, x). For instance, if we consider two versions of Beethoven’s Grosse Fuge, say, by the Juilliard Quartet and the Berg Quartet, we may have mixed and conﬂicting feelings. We may prefer to some extent the Berg version because of its energy but have also some preference for the Juilliard because of its poetry. This will entail that some value α ∈]0, 1] be given to the preference for the Berg version over the Juilliard, but also that some value β ∈]0, 1] be given to the preference for the Juilliard version over the Berg.

Moreover, on the basis of this example, it might seem strange to describe fuzziness by assigning a precise number. The logician Alasdair Urquhart (2001) has rightly observed :

One immediate objection that presents itself to this line of approach is the extremely artiﬁcial nature of the attaching of precise numerical values to sentences like ‘73 is a large number’ or ‘Picasso’s Guernica is beautiful’. In fact, it seems plausible to say that the nature of vague predicates precludes attaching precise numerical values just as much as it precludes attaching precise classical truth values.

One way to avoid this diﬃculty is to replace [0, 1] ordered by ≥ by some set of qualitative elements ordered by a complete preorder. For instance, to give some intuition in the context of preference, the elements could be interpreted as being ‘not at all’, ‘insignif- icantly’, ‘a little’, ‘mildly’, ‘much’, ‘very much’, ‘deﬁnitely’, etc. We will consider both cases in the following subsections.

2.1. Fuzzy preferences: numerical values. We will stick in this chapter to the notions and properties used in the fuzzy aggregation literature. There are many notions about transitivity and choice which will not be introduced (see, for instance, Barrett, Pattanaik and Salles (1990), Basu, Deb and Pattanaik (1992), Dasgupta and Deb ((1991), (1996), (2001)), Salles (1998)). SOCIAL CHOICE WITH FUZZY PREFERENCES 3

Let X be the set of alternatives with #X ≥ 3.

We will consider two types of fuzzy binary relations. Strict binary relations as described here were introduced in Barrett, Pattanaik and Salles (1986), hereafter denoted by BPS, and weak binary relations were introduced by Dutta (1987).

Deﬁnition 1. A fuzzy binary relation over X is a function h : X × X → [0, 1].

Deﬁnition 2. A fuzzy binary relation p is a BPS-fuzzy strict preference if for all distinct x, y, z ∈ X, (i) p(x, x) = 0 (ii) p(x, y) = 1 ⇒ p(y, x) = 0 (iii) p(x, y) > 0 and p(y, z) > 0 ⇒ p(x, z) > 0.

Deﬁnition 3. A BPS-fuzzy strict preference p is BPS-complete if for all distinct x, y ∈ X, p(x, y) > 0 or p(y, x) > 0.

p(x, y) can be interpreted as the degree of intensity with which x is preferred to y (or, see above, the degree of truth that x is preferred to y). (i) expresses that we consider strict preferences. (ii) means that when strict preferences are deﬁnite (in some sense non-fuzzy), they are asymmetric. (iii) is a rather mild transitivity property. It is now well known that many transitivity concepts are available for fuzzy binary relations (Das- gupta and Deb, 1996). In particular, the most widely used concept, max-min transitivity, which states that given any x, y, z ∈ X, p(x, z) ≥ min(p(x, y), p(y, z)) is stronger than (iii).

Subramanian (1987) uses two diﬀerent fuzzy strict preferences one of which is a variant of BPS-fuzzy strict preferences.

S1 Definition 4. A fuzzy strict preference p is a S1-fuzzy strict preference if it is a BPS-fuzzy strict preference for which for all distinct x, y, z ∈ X, pS1 (x, y) = pS1 (y, x) = pS1 (y, z) = pS1 (z, y) = 0 ⇒ pS1 (x, z) = pS1 (z, x) = 0. S2 A fuzzy strict preference p is a S2-fuzzy strict preference if it satisfies properties (i) and (ii) of Definition 2, and if S2 S2 S2 S2 for all distinct x1, x2, ..., xk ∈ X, p (x1, x2) > p (x2, x1) and p (x2, x3) > p (x3, x2) S2 S2 S2 S2 and ... and p (xk−1, xk) > p (xk, xk−1) ⇒ ¬(p (xk, x1) = 1 and p (x1, xk) = 0).

Deﬁnition 5. A fuzzy binary relation r is a fuzzy weak preference if it is reﬂexive, i.e., if for all x ∈ X, r(x, x) = 1.

The basic idea underlying the concept of fuzzy weak preference is that it will be possible to derive from it two components, viz. a symmetric component, the fuzzy indiﬀerence, i 1, and more importantly a strict component which is in some fuzzy sense asymmetric, p. All the numerical fuzzy weak preferences we are considering in this chapter are connected (or complete) with the following meaning.

Deﬁnition 6. A fuzzy weak preference r is connected if for all x, y ∈ X, r(x, y) + r(y, x) ≥ 1.

We will present three decompositions of a fuzzy weak preference introduced respectively by Dutta (1987), Banerjee (1994), and Richardson (1998) and Dasgupta and Deb (1999). Theoretical results about these decompositions can be found in the cited papers and in

1Although the letter i is also used as the generic letter for individuals, there will clearly be no confusion possible. 4 RICHARD BARRETT AND MAURICE SALLES

Dasgupta and Deb (2001). We will not present these results here since they are not directly connected with the topic of the chapter.

Deﬁnition 7. A fuzzy weak preference rD is a D-fuzzy weak preference if for all x, y ∈ X, iD(x, y) = min(rD(x, y), rD(y, x)) and rD(x, y) if rD(x, y) > rD(y, x) (2.1) pD(x, y) = 0 otherwise Banerjee objects to this decomposition on the basis that if iD(x, y) > 0, pD(x, y) should be less than rD(x, y). It seems strange that when rD(x, y) = 1 and rD(y, x) = 0.999, and when rD(x, y) = 1 and rD(y, x) = 0, pD(x, y) has the same value (1). Richardson adds that the discontinuity of pD seems also rather unreasonable. If rD(x, y) = 1 and rD(y, x) = 0.999, pD(x, y) = 1, and if rD(x, y) = rD(y, x) = 1, pD(x, y) = 0.

Deﬁnition 8. A fuzzy weak preference rB is a B-fuzzy weak preference if for all x, y ∈ X, iB(x, y) = min(rB(x, y), rB(y, x)) and pB(x, y) = 1 − rB(y, x).

Richardson discusses this decomposition and the rˆoleof a property of strong connectedness. In particular, he notes that pB(x, y) has the same value when rB(x, y) = 1 and rB(y, x) = 0.999, and when rB(x, y) = 0.001 and rB(y, x) = 0.999.

Deﬁnition 9. A fuzzy weak preference r is strongly connected if for all x, y ∈ X, max(r(x, y), r(y, x)) = 1.

2 Deﬁnition 10. A fuzzy weak preference rRD is a RD2-fuzzy weak preference if for all 2 2 2 2 2 x, y ∈ X, iRD (x, y) = min(rRD (x, y), rRD (y, x)) and pRD (x, y) = max(rRD (x, y) − 2 rRD (y, x), 0).

In these deﬁnitions D is for Dutta (1987), B is for Banerjee (1994) and RD2 is for both Richardson (1998) and Dasgupta and Deb (1999). We will now introduce several transitivity properties for fuzzy weak preferences.

Deﬁnition 11. A fuzzy binary relation r is (i) max-min transitive if for all x, y, z ∈ X, r(x, z) ≥ min(r(x, y), r(y, z)), (ii) max-δ transitive if for all x, y, z ∈ X, r(x, z) ≥ r(x, y) + r(y, z) − 1, (iii) exactly transitive if for all x, y, z ∈ X, r(x, y) = 1 and r(y, z) = 1 ⇒ r(x, z) = 1, (iv) weakly max-min transitive if for all x, y, z ∈ X, if r(x, y) ≥ r(y, x) and r(y, z) ≥ r(z, y), then r(x, z) ≥ min(r(x, y), r(y, z)).

Max-min transitivity implies max-δ transitivity which implies exact transitivity. Also, it is obvious that max-min transitivity implies weak max-min transitivity (see Dasgupta and Deb (1996) for further results). Based on these four transitivities, we will define various social welfare functions in the next section. As mentioned above, there are many transitivity concepts for fuzzy binary relations (in fact, obviously an infinity). A basic requirement is that, when applied to values restricted to be 0 or 1 (i.e., to crisp relations), one must recover the standard notions of transitivity. For instance, if we consider (iii) of Definition 2, we have : p(x, y) = 1 and p(y, z) = 1 ⇒ p(x, z) = 1, which is the transitivity of the standard (strict) preference. But if we consider the following definition : for all distinct x, y, z ∈ X, p(x, y) > .05 and p(y, z) > .01 ⇒ p(x, z) = 1 we have a transitivity notion for a fuzzy binary relation which is compatible with its crisp counterpart without being intuitively convincing. It is furthermore independent of (iii) of Definition 2. Of course, if, moreover, the transitivity properties are not independent, then, all other things being equal, for impossibility results, the weakest notion is the best, and for possibility results, the strongest notion the best. SOCIAL CHOICE WITH FUZZY PREFERENCES 5

2.2. Fuzzy preferences: qualitative values. We mentioned above that assigning precise numbers to elements to describe vagueness could seem paradoxical. Goguen (1967) and Basu, Deb and Pattanaik (1992) have proposed assigning some qualitative value, the set of these values being subject to some binary relation. We will follow here Barrett, Pattanaik and Salles (1992) and consider fuzzy strict preferences where fuzziness is given by elements in a finite set L completely preordered by a relation . This is, of course, very similar to Goguen’s L-fuzzy sets, the only difference being that in Goguen is a linear order, i.e., an anti-symmetric complete preorder. This means that, with a non-anti-symmetric complete preorder, there might be a non-unique way to define a degree of fuzziness (for instance ‘a little’ and ‘mildly’ can express the same fuzziness, though they are different elements of L).

Let L be a ﬁnite set and a complete preorder on L with a unique -maximum, de- ? noted d , and a unique -minimum, denoted d?.

Deﬁnition 12. An ordinally fuzzy binary relation H is a function H : X × X → L.

Deﬁnition 13. An ordinally fuzzy binary relation P is a BPS-ordinally fuzzy strict preference if for all distinct x, y, z ∈ X, (i) P (x, x) = d? ? (ii) P (x, y) = d ⇒ P (y, x) = d? (iii) P (x, y) = d? ⇒ P (x, z) P (y, z), and P (y, z) = d? ⇒ P (x, z) P (x, y).

We will introduce a variety of transitivity conditions. As previously, it might be difficult to say which condition is the most appropriate. It can depend on the context. However we impose with (iii) a sort of transitivity condition which seems as compelling as (i) (exact irreflexivity) and (ii) (exact asymmetry). According to (iii) if x is definitely (or exactly) better than y, then x must ‘fare as well’ against z as y against z (in terms of preference in favour), and if y is definitely better than z, then x must ‘fare as well’ against z as against y. If one definitely prefers Bartok’s concerto for orchestra to Gorecki’s third symphony, the degree of his preference for Bartok’s concerto over Dutilleux’s first symphony (which may be ‘mild’) must be ‘at least as strong as’ the degree of his preference for Gorecki’s symphony over Dutilleux’s symphony (which may be null).

Deﬁnition 14. Let P be a BPS-ordinally fuzzy strict preference. P is (i) weakly max-min transitive if for all x, y, z ∈ X, P (x, y) P (y, x) and P (y, z) P (z, y) ⇒ P (x, z) P (x, y) or P (x, z) P (y, z), (ii) quasi-transitive if for all x, y, z ∈ X, P (x, y) P (y, x) and P (y, z) P (z, y) ⇒ P (x, z) P (z, x), (iii) acyclical if there is no ﬁnite set {x1, ..., xk} ⊆ X (k > 1) such that P (x1, x2) P (x2, x1) and ... and P (xk−1, xk) P (xk, xk−1) and P (xk, x1) P (x1, xk), (iv) simply transitive if for all x, y z ∈ X,(P (x, y) d? and P (y, x) = d?) and (P (y, z) d? and P (z, y) = d?) ⇒ P (x, z) d? and P (z, x) = d?.

The following example justifies the choice of these transitivity properties as compared with others. Consider three options: a sum of money m, then m+δ (δ > 0) and x which is ? 0 ? unspecified. Suppose P (m + δ, m) = d , P (m, x) = d, P (x, m + δ) = d , with d d d? ? 0 and d d d?. If we consider the ordinal version of max-min transitivity, i.e., if for all x, y z ∈ X, P (x, z) P (x, y) or P (x, z) P (y, z), we should obtain P (m, m + δ) d or 0 ? P (m, m + δ) d . Since P (m + δ, m) = d , by (ii) in Definition 13, P (m, m + δ) = d?, a contradiction. The same is true if we consider the ordinal version of (iii) in Definition 2, i.e., for all x, y z ∈ X, P (x, y) d? and P (y, z) d? ⇒ P (x, z) d?. However, this example is compatible with our four transitivity properties. 6 RICHARD BARRETT AND MAURICE SALLES

3. aggregation of fuzzy preferences: Arrovian theorems We will introduce in all the cases deﬁned above aggregation procedures and properties which are essentially the fuzzy replicates of Arrow’s conditons (see Arrow (1963), Sen (1970)).

3.1. The case of numerical values. Let N = {1, ..., n} be a ﬁnite set of individuals (n ≥ 2).

Deﬁnition 15. A fuzzy aggregation function is a function that associates a social fuzzy binary relation over X, denoted hS, to an n-list of individual fuzzy binary relations over X, denoted (h1, ..., hi, ..., hn).

hi(x, y) can be interpreted as the degree of intensity with which individual i prefers (weakly or strictly) x to y (or the degree of conﬁdence we have that i prefers (weakly or strictly) x to y).

Deﬁnition 16. Let f be a fuzzy aggregation function, hi, hS, etc., be fuzzy binary relations, pi, pS, etc., be BPS-fuzzy strict preferences, or strict components of ri, rS of any type (i.e. D, B or RD2), etc. f satisﬁes 0 0 FI (fuzzy independence of irrelevant alternatives) if for all n-lists (h1, ..., hn), (h1, ..., hn) 0 0 and all distinct x, y ∈ X, hi(x, y) = hi(x, y) and hi(y, x) = hi(y, x) for every i ∈ N 0 0 0 implies hS(x, y) = hS(x, y) and hS(y, x) = hS(y, x), where hS = f(h1, ..., hn) and hS = 0 0 f(h1, ..., hn); FPC (fuzzy Pareto criterion) if for all (h1, ..., hn), all distinct x, y ∈ X, pS(x, y) ≥ 2 minipi(x, y), where hS = f(h1, ..., hn); and 0 0 FPR (fuzzy positive responsiveness) if for all (r1, ..., rn), (r1, ..., rn) and all distinct 0 0 x, y ∈ X, ri = ri for all i 6= j, rS(x, y) = rS(y, x), and (pj(x, y) = 0 and pj(x, y) > 0) or 0 0 (pj(y, x) > 0 and pj(y, x) = 0) ⇒ pS(x, y) > 0.

FI is the natural counterpart of Arrow’s independence of irrelevant alternatives and FPC means that if every individual prefers x to y with at least degree t , then the society must reﬂect this unanimity.

0 0 Let ABPS be the set of BPS-fuzzy strict preferences and ABPS ⊆ ABPS, ABPS 6= ∅.

Deﬁnition 17. A BPS-fuzzy social welfare function is a fuzzy aggregation function 0n f : ABPS → ABPS.

0n Deﬁnition 18. Let f : ABPS → ABPS and Jf = {(t1, t2) ∈ [0, 1] × [0, 1]: for some 0 p ∈ ABPS and some distinct a, b ∈ X, p(a, b) = t1 and p(b, a) = t2}. f is said to have a non-narrow domain for distinct x, y, z ∈ X if 0 for all (t1, t2) ∈ Jf , there exists p ∈ ABPS such that p(x, y) = p(x, z) = 1 and p(y, z) = 0 0 0 0 t1 and p(z, y) = t2, and also there exists p ∈ ABPS such that p (y, x) = p (z, x) = 1 and 0 0 p (y, z) = t1 and p (z, y) = t2, and for all (t1, t2) ∈ [0, 1] × [0, 1] for which (t1, 0), (t2, 0) ∈ Jf and t2 ≥ t1, there exists p ∈ 0 ABPS such that p(x, y) = t1, p(y, z) = 1, p(x, z) = t2 and p(y, x) = p(z, y) = p(z, x) = 0, 0 0 0 0 0 and also there exists p ∈ ABPS such that p (x, y) = 1, p (y, z) = t1, p (x, z) = t2 and p0(y, x) = p0(z, y) = p0(z, x) = 0.

0 Of course, if ABPS = ABPS, it can be easily seen that the condition deﬁning a non- narrow domain is satisﬁed for all distinct x, y, z ∈ X. This condition is weaker than a

2 (h1, ..., hn) is either (p1, ..., pn) in the BPS framework or (r1, ..., rn), with hS being respectively pS or rS . SOCIAL CHOICE WITH FUZZY PREFERENCES 7

0 universality condition requiring that ABPS = ABPS and is suﬃcient to obtain the following theorems. A coalition is a non-empty subset of N.

0n Theorem 1. Let f : ABPS → ABPS be a BPS-fuzzy social welfare function satisfying FI, FPC and having a non-narrow domain for all distinct x, y, z ∈ X. Then there exists a unique coalition C such that 0n for all distinct x, y ∈ X and all (p1, ..., pn) ∈ ABPS, if pi(x, y) > 0 and pi(y, x) = 0 for every i ∈ C, then pS(x, y) > 0, where pS = f(p1, ..., pn); and 0n for all distinct x, y ∈ X and all (p1, ..., pn) ∈ ABPS, if for some j ∈ C, pj(x, y) > 0 and pj(y, x) = 0, then pS(y, x) = 0, where pS = f(p1, ..., pn).

This theorem is reminiscent of Gibbard’s oligarchy theorem (Gibbard (1969)). If the individuals in coalition C share some agreement in their preferences, they can exert some positive (fuzzy) power. Furthermore, each individual in the coalition has some fuzzy veto power.

One can verify that the fuzzy aggregation function given by pS(x, y) = minipi(x, y) is a BPS-fuzzy social welfare function. This does not contradict Theorem 1. In this case the unique coalition C is the entire N.

0n Theorem 2. Consider the further requirement that if pS ∈ f(ABPS), then pS is BPS- complete. Then #C = 1, i.e., the coalition C shrinks: there exists an individual i ∈ N such that for all distinct x, y ∈ X and all (p1, ...pn) ∈ 0n ABPS, if pi(x, y) > 0 and pi(y, x) = 0, then pS(x, y) > 0 and pS(y, x) = 0, where pS = f(p1, ..., pn).

This theorem is a fuzzy analog of Arrow’s theorem (Arrow (1963)) and the individual of Theorem 2 could be considered a BPS-fuzzy dictator.

We will now consider fuzzy weak preferences.

Let AD1 be the set of D-fuzzy weak preferences which are max-min transitive.

Deﬁnition 19. A D1-fuzzy social welfare function is a fuzzy aggregation function f : n → . AD1 AD1

For such functions, one obtains a result similar to Theorem 1.

Theorem 3. Let f : n → be a D -fuzzy social welfare function satisfying FI AD1 AD1 1 and FPC. Then there exists a unique coalition C such that for all distinct x, y ∈ X and all (rD, ..., rD) ∈ n , if pD(x, y) > 0 for every i ∈ C, 1 n AD1 i D D D D then pS (x, y) > 0, where rS = f(r1 , ..., rn ); and for all distinct x, y ∈ X and all (rD, ..., rD) ∈ n , if for some j ∈ C, pD(x, y) > 0, 1 n AD1 j D D D D then pS (y, x) = 0, where rS = f(r1 , ..., rn ).

Although Dutta (1987) provides an example showing that Theorem 2 cannot be directly extended (take rS(x, x) = 1, and for x 6= y rS(x, y) = 1 if for all i ∈ N ri(x, y) > ri(y, x), and rS(x, y) = α ∈]1/2, 1[ otherwise), a result similar to Theorem 2 is possible if one further assumes that f satisfies the positive responsiveness conditon defined above (Definition 16).

Theorem 4. Let n ≥ 3 and f : n → be a D -fuzzy social welfare function AD1 AD1 1 satisfying FI, FPC and FPR. Then there exists an individual i ∈ N such that for all distinct x, y ∈ X and all (rD, ...rD) ∈ n , if pD(x, y) > 0, then pD(x, y) > 0, where 1 n AD1 i S 8 RICHARD BARRETT AND MAURICE SALLES

D D D rS = f(r1 , ..., rn ).

The result is reminiscent of a result of Mas-Colell and Sonnenschein (1972). The individual shown to exist is a sort of fuzzy dictator. However, if max-min transitivity is replaced by max-δ transitivity, the kind of impossibility of Theorem 4 vanishes.

Let AD2 be the set of D-fuzzy weak preferences which are max-δ transitive.

Deﬁnition 20. A D2-fuzzy social welfare function is a fuzzy aggregation function f : n → . AD2 AD2

Theorem 5. For all (rD, ..., rD) ∈ n and all x, y ∈ X, let r (x, y) = (1/n)Σ rD(x, y).3 1 n AD2 S i i Then this function is a D2-fuzzy social welfare function satisfying FI, FPC and FPR for D D which there is no individual i ∈ N such that for all distinct x, y ∈ X and all (r1 , ..., rn ) ∈ n , if pD(x, y) > 0, then pD(x, y) > 0, where rD = f(rD, ..., rD). AD2 i S S 1 n

In fact this rule is also obviously anonymous, with anonymity deﬁned in the usual way as symmetry over individuals (incidentally, it is also symmetric over options).

Banerjee (1994) shows that on substituting fuzzy weak preferences rB, Theorem 5 is no longer true and a theorem similar to Theorem 4 is obtained. However, as Richardson (1998) observes, Banerjee uses the hidden fact that two of the suﬃcient conditions to obtain his decomposition imply that the fuzzy weak preferences rB are strongly connected. One of these two conditions is in fact imposed by all the mentioned contributors. It is the sort of fuzzy asymmetry mentioned above. It says that the strict component p must 2 satisfy p(x, y) > 0 ⇒ p(y, x) = 0. If it is obviously the case for rD and rRD , it is not for rB. However, by adding strong connectedness, it also becomes true for rB.

Let AB be the set of B-fuzzy weak preferences which are strongly connected and max-δ transitive.

Deﬁnition 21. A B-fuzzy social welfare function is a fuzzy aggregation function n f : AB → AB.

n Theorem 6. Let f : AB → AB be a B-fuzzy social welfare function satisfying FI and FPC. Then there exists an individual i ∈ N such that for all distinct x, y ∈ X and all B B n B B B B B (r1 , ..., rn ) ∈ AB, if pi (x, y) > 0, then pS (x, y) > 0, where rS = f(r1 , ..., rn ).

2 We will now consider the third decomposition, i.e., fuzzy weak preferences rRD .

2 Let ARD2 be the set of RD -fuzzy weak preferences which are strongly connected and exactly transitive.

Deﬁnition 22. A RD2-fuzzy social welfare function is a fuzzy aggregation function n f : ARD2 → ARD2 .

We may note that exact transitivity is a very weak condition, so the following theorem due to Richardson is quite interesting even though strong connectedness may appear as rather constraining.

n 2 Theorem 7. Let f : ARD2 → ARD2 be a RD -fuzzy social welfare function satisfying FI and FPC. Then there exists an individual i ∈ N such that for all distinct

3This function is sometimes called the mean rule; for extended studies see Garc´ıa-Lapresta and Lla- mazares (2000) and Ovchinnikov (1991). SOCIAL CHOICE WITH FUZZY PREFERENCES 9

RD2 RD2 n RD2 RD2 x, y ∈ X and all (r1 , ..., rn ) ∈ ARD2 , if pi (x, y) > 0, then pS (x, y) > 0, where RD2 RD2 RD2 rS = f(r1 , ..., rn ).

However, if we assume that the RD2-fuzzy weak preferences are only max-δ transitive, we get again a kind of possibility result. 2 Let BRD2 be the set of RD -fuzzy weak preferences which are max-δ transitive.

2 Deﬁnition 23. A RDδ -fuzzy social welfare function is a fuzzy aggregation function n f : BRD2 → BRD2 .

RD2 RD2 n Theorem 8. For all (r1 , ..., rn ) ∈ BRD2 and all x, y ∈ X, let rS(x, y) = RD2 2 (1/n)Σiri (x, y). Then this function is a RDδ -fuzzy social welfare function satisfying FI, FPC and FPR for which there is no individual i ∈ N such that for all distinct RD2 RD2 n RD2 RD2 x, y ∈ X and all (r1 , ..., rn ) ∈ BRD2 , if pi (x, y) > 0, then pS (x, y) > 0, where RD2 RD2 RD2 rS = f(r1 , ..., rn ).

Weak max-min transitivity is found by Dasgupta and Deb ((1996), (2001)) to perform well in permitting non-trivial fuzzy preferences and in preventing cycles of strict preferences. By using RD2 decomposition they obtain again a rather negative result, showing the existence of an individual having disproportionate power.

2 Let CRD2 be the set of RD -fuzzy weak preferences which are weakly max-min transitive.

2 Deﬁnition 24. A RDw-fuzzy social welfare function is a fuzzy aggregation function n f : CRD2 → CRD2 .

n 2 Theorem 9. Let f : CRD2 → CRD2 be a RDw-fuzzy social welfare function satisfying FI and FPC. Then there exists an individual i ∈ N such that for all distinct RD2 RD2 n RD2 RD2 x, y ∈ X and all (r1 , ..., rn ) ∈ CRD2 , if pi (x, y) = 1, then pS (x, y) > 0, where RD2 RD2 RD2 rS = f(r1 , ..., rn ).

The kind of fuzzy dictator we have here, called a weak dictator by Dasgupta and Deb (1999), exerts power only in case he or she exactly prefers one option to another.

3.2. The case of qualitative values. This subsection will be entirely based on Barrett, Pattanaik and Salles (1992). Fuzzy aggregation in an ordinal framework has rarely been explored. The only other work we know is Barrett and Pattanaik (1990) where Barrett, Pattanaik and Salles (1992) is extended in characterizing rank-based aggregation rules such as the median rule.

Deﬁnition 25. An ordinally fuzzy aggregation function is a function that associates a social ordinally fuzzy binary relation over X, denoted HS, to an n-list of individual ordinally fuzzy binary relations over X, denoted (H1, ..., Hi, ..., Hn).

Deﬁnition 26. Let f be an ordinally fuzzy aggregation function, Hi, HS, etc., be ordinally fuzzy binary relations, Pi, PS, etc., be BPS-ordinally fuzzy strict preferences. f satisﬁes OFI (ordinally fuzzy independence of irrelevant alternatives) if for all n-lists (H1, ..., Hn), 0 0 0 0 (H1, ..., Hn) and all x, y ∈ X, Hi(x, y) ∼ Hi(x, y) and Hi(y, x) ∼ Hi(y, x) for every i ∈ N 0 0 implies HS(x, y) ∼ HS(x, y) and HS(y, x) ∼ HS(y, x); and OFU (ordinally fuzzy unanimity) if for all n-lists (P1, ..., Pn) and all x, y ∈ X, there exists i ∈ N such that Pi(x, y) PS(x, y) and there exists j ∈ N such that PS(x, y) 10 RICHARD BARRETT AND MAURICE SALLES

Pj(x, y).

Condition OFU is a sort of Pareto criterion. Consider (P1, ..., Pn) and x, y ∈ X. The 4 set {max`P`(x, y)} (` = 1, ...n) contains some element α. Similarly, {min`P`(x, y)} contains some element β. The condition means that the ordinally fuzzy strict social preference PS(x, y) must be ‘between’ α and β or ‘at the same level’ as α or β, i.e., α PS(x, y) β.

Let O be the set of BPS-ordinally fuzzy strict preferences and Ow, Oq, Oa, and Os be respectively the set of BPS-ordinally fuzzy strict preferences which are weakly max-min transitive, quasi-transitive, acyclical, and simply transitive.

Theorem 10. Ow ⊆ Oq ⊆ Oa.

We will now introduce several ordinally fuzzy aggregation functions distinguished according to the set in which these functions take their values. The domains of these functions are identical, viz., the Cartesian product of Ow (or, of course, the Cartesian product of any superset of Ow, though it will not be indicated).

Deﬁnition 27. A w-(respectively q-, a-, s-)ordinally fuzzy social welfare function is an n n n ordinally fuzzy social welfare function f : Ow → Ow (respectively Ow → Oq, Ow → Oa, n Ow → Os).

According to the diﬀerent transitivity properties and other conditions imposed, we obtain the ﬁve following results.

n Theorem 11. Let #X ≥ n and let f : Ow → Oa be an a-ordinally fuzzy social welfare function satisfying OFI and OFU. Let d ∈ L, d d?. Then there exists an individual n j such that for all x, y ∈ X and for all (P1, ..., Pn) ∈ Ow, Pj(x, y) d Pj(y, x) and d Pi(y, x) for all i ∈ N − {j} ⇒ PS(x, y) PS(y, x), where PS = f(P1, ..., Pn).

This is clearly an expression of veto power. This sort of veto power can even be strength- ened if the number of options is increased.

n Theorem 12. Let #X ≥ 2n and let f : Ow → Oa be an a-ordinally fuzzy social welfare function satisfying OFI and OFU. Let d ∈ L, d d?. Then there exists an individual n j such that for all x, y ∈ X and for all (P1, ..., Pn) ∈ Ow, Pj(x, y) d Pj(y, x) ⇒ PS(x, y) PS(y, x), where PS = f(P1, ..., Pn).

It is possible to extend these two theorems when no restriction is imposed on the number of elements in X. Regarding Theorem 11 if dn/#Xe is the smallest integer ≥ n/#X, N can be partitioned into at most #X coalitions of size ≤ dn/#Xe. Then, the kind of veto power assigned to individual j, will be assigned to some coalition belonging to the partition. For Theorem 12, it is suﬃcient to replace dn/#Xe by d2n/#Xe.

We will consider now the case of quasi-transitivity.

n Theorem 13. Let f : Ow → Oq be a q-ordinally fuzzy social welfare function satisfying OFI and OFU. Let d ∈ L, d d?. Then there exists a coalition C such that: n for all x, y ∈ X and for all (P1, ..., Pn) ∈ Ow, Pi(x, y) d Pi(y, x) for all i ∈ C ⇒ PS(x, y) PS(y, x), where PS = f(P1, ..., Pn); and

4Since is a complete preorder, we have not excluded the possibility that two diﬀerent elements essentially represent the same ‘degree of preference’ (for instance ‘a little’ and ‘mildly’). Then {max`P`(x, y)} may contain more than one element. SOCIAL CHOICE WITH FUZZY PREFERENCES 11

n for all i ∈ C, for all x, y ∈ X and for all (P1, ..., Pn) ∈ Ow, Pi(x, y) d Pi(y, x) ⇒ PS(x, y) PS(y, x), where PS = f(P1, ..., Pn).

Replacing Oq by Ow, we obtain the following theorem.

n Theorem 14. Let f : Ow → Ow be a w-ordinally fuzzy social welfare function satisfying OFI and OFU. Let d ∈ L, d d?. Then there exists a coalition C such that: n for all x, y ∈ X and for all (P1, ..., Pn) ∈ Ow, Pi(x, y) d Pi(y, x) for all i ∈ C ⇒ PS(x, y) d PS(y, x), where PS = f(P1, ..., Pn); and n for all i ∈ C, for all x, y ∈ X and for all (P1, ..., Pn) ∈ Ow, Pi(x, y) d Pi(y, x) ⇒ PS(x, y) d or d PS(y, x), where PS = f(P1, ..., Pn).

Finally, we consider the case of simple transitivity.

n Theorem 15. Let f : Ow → Os be an s-ordinally fuzzy social welfare function satisfying OFI and OFU. Then there exists a coalition C such that: n for all x, y ∈ X and for all (P1, ..., Pn) ∈ Ow, Pi(x, y) d? and Pi(y, x) = d? for all i ∈ C ⇒ PS(x, y) d? and PS(y, x) = d?, where PS = f(P1, ..., Pn); and n for all i ∈ C, for all x, y ∈ X and for all (P1, ..., Pn) ∈ Ow, Pi(x, y) d? and Pi(y, x) = d? ⇒ PS(x, y) d? or PS(y, x) = d?, where PS = f(P1, ..., Pn).

By Theorem 10, one can replace appropriately the transitivity properties in Theorems 11, 12 and 13. Theorem 15 can be compared to Theorem 1. The result of Theorem 15 is slightly weaker regarding the kind of veto power, since we have PS(x, y) d? or PS(y, x) = d? rather than pS(y, x) = 0.

4. other aspects In this section, we will present two aspects which are heretofore less developed than what can be called Arrovian aspects, viz., fuzzy versions of Sen’s impossibility of a Paretian liberal and considerations of some economic types of restriction on fuzzy preferences.

4.1. Aggregation of fuzzy preferences and Sen’s impossibility theorem. This subsection is essentially based upon Subramanian (1987).

Sen’s impossibility theorem demonstrates that for some class of aggregation functions (generally called social decision functions) that includes Arrovian social welfare functions, given a sufficiently large domain, there is an inconsistency between unanimity—or the weak Pareto principle—(whenever every individual prefers alternative a to alternative b, so does the society) and a condition called minimal liberalism (there are at least two individuals i and j and for each of them two alternatives, ai, bi for i (resp. aj, bj for j), such that the (strict) preference of i (resp. j) over his alternatives is reflected by the social (strict) preference over these alternatives. The intuition is that the alternatives ai, bi for i (resp. aj, bj for j) belong to i’s (resp. j’s) personal sphere, or, more precisely, differ on characteristics concerning i (resp. j) only. This condition may appear as being very strong and giving too much power to these two individuals. Rather than describing individual liberty, it can be interpreted as some kind of local dictatorship (Salles (2000)). In Subramanian’s fuzzy version, the condition is weakened. Whenever i (resp. j) exactly prefers one of his two alternatives to the other, say ai to bi, then the degree of the fuzzy social preference of ai over bi must be greater than (or at least as great as) the degree of the fuzzy social preference of bi over ai.

Let S1 be the set of S1-fuzzy strict preferences, S2 the set of S2-fuzzy strict preferences, S1e the subset of S1 made up of all exact S1-fuzzy strict preferences (those for which the only possible values are 0 or 1), S2e the subset of S2 made up of all exact S2-fuzzy strict 12 RICHARD BARRETT AND MAURICE SALLES preferences. (See Deﬁnition 4.)

Deﬁnition 28. A S1-fuzzy social welfare function is a fuzzy aggregation function f : n n S1 → S2.A S1e-fuzzy social welfare function is a fuzzy aggregation function f : S1e → S2. 0 n A S1e-fuzzy social welfare function is a fuzzy aggregation function f : S1e → S1.

In the following definition, f will be any of the fuzzy aggregation functions defined in Definition 28, and we will use generically (p1, ..., pn) for any n-list in the domains of these functions.

Definition 29. Let f be any of the fuzzy aggregation functions of Definition 28. f satisfies SFPC (S-fuzzy Pareto criterion) if for all (p1, ..., pn), all distinct x, y ∈ X, pi(x, y) = 1 and pi(y, x) = 0 for all i ∈ N ⇒ pS(x, y) = 1 and pS(y, x) = 0, where pS = f(p1, ..., pn); and FML1 (fuzzy minimal liberalism-1) if there exist two individuals i, j ∈ N and for each of them two options, ai and bi for i and aj and bj for j, such that for all (p1, ..., pn), pi(ai, bi) = 1 and pi(bi, ai) = 0 ⇒ pS(ai, bi) > pS(bi, ai), and for all (p1, ..., pn), pi(bi, ai) = 1 and pi(ai, bi) = 0 ⇒ pS(bi, ai) > pS(ai, bi); and for all (p1, ..., pn), pj(aj, bj) = 1 and pj(bj, aj) = 0 ⇒ pS(aj, bj) > pS(bj, aj), and for all (p1, ..., pn), pj(bj, aj) = 1 and pj(aj, bj) = 0 ⇒ pS(bj, aj) > pS(aj, bj); and FML2 (fuzzy minimal liberalism-2) if there exist two individuals i, j ∈ N and for each of them two options, ai and bi for i and aj and bj for j, such that for all (p1, ..., pn), pi(ai, bi) = 1 and pi(bi, ai) = 0 ⇒ pS(ai, bi) ≥ pS(bi, ai), and for all (p1, ..., pn), pi(bi, ai) = 1 and pi(ai, bi) = 0 ⇒ pS(bi, ai) ≥ pS(ai, bi); and for all (p1, ..., pn), pj(aj, bj) = 1 and pj(bj, aj) = 0 ⇒ pS(aj, bj) ≥ pS(bj, aj), and for all (p1, ..., pn), pj(bj, aj) = 1 and pj(aj, bj) = 0 ⇒ pS(bj, aj) ≥ pS(aj, bj).

It should be noted that both liberalism conditions are based on individual exact preferences. The ﬁrst one is interpreted by Subramanian as the fuzzy counterpart of the condition due to Sen ((1970), (1970a)) and the second of the version due to Karni (1978). Also the Pareto criterion (unanimity) is based on individual and social exact preferences.

n Theorem 16. There does not exist a function f : S1e → S2 satisfying conditions SFPC and FML1.

n Theorem 17. There does not exist a function f : S1e → S1 satisfying conditions SFPC and FML2.

Theorem 18. The fuzzy aggregation function f deﬁned by n for all x, y ∈ X, all (p1, ..., pn) ∈ S1 , pS(x, y) = minipi(x, y), where pS = f(p1, ..., pn), is a S1-fuzzy social welfare function satisfying SFPC and FML2.

The crucial rˆoleof exact preferences in these theorems must be underlined. Exact preferences are the only preferences in the domains for Theorems 16 and 17, and furthermore, individual preferences are exact in the deﬁnitions of the Pareto condition (SFPC) and the fuzzy versions of minimal liberalism conditions.

Dimitrov (2004) uses intuitionistic fuzzy sets introduced by Atanassov (1999). Rather than associating a (unique) number to an ordered pair (x, y), he associates two numbers, the ﬁrst one expressing the degree to which x is preferred to y and the second the degree to which x is not preferred to y, requiring that the sum of these numbers be ≤ 1. He consid- ers fuzzy weak preferences with a decomposition `ala Dutta and obtains a possibility result. SOCIAL CHOICE WITH FUZZY PREFERENCES 13

4.2. Aggregation of fuzzy preferences and economic environments. When social choice theory started its modern development in the 1940’s, Black (1958) introduced a condition (in a kind of geometric way), called single-peakedness, restricting the individual preferences. In the exact case, if the individual preferences given by complete preorders are single-peaked, majority rule is a social welfare function (given some mild condition on the number of individuals having these preferences). A number of developments took place from the 1960’s. They are excellently surveyed in Gaertner (2001, 2002). Among the restrictive conditions on individual preferences, those used in standard microeconomic theory are particularly important. For instance, consider exchange economies. Since equilibrium redistributions are Pareto-optimal, it seems crucial to be able to rank these redistributions on the basis of an aggregation function that satisfies some properties related to ethical and social justice considerations. A major difficulty arises about these considerations because individuals have preferences over their consumption sets but not on redistributions. This can be resolved by assuming selfishness and by identifying an individual’s preferences over the redistributions with this individual’s preferences over his or her individual bundles. In the case of (pure) public goods, this difficulty disappears as individuals and society have their preferences on the same set of alternatives, which is generally taken to be the positive orthant of a Euclidean space. Then an individual’s preference is generally given by a complete preorder which is, furthermore, monotonic, continuous and (strictly) convex. In a fundamental paper that started the literature on aggregation in an economic environment, Kalai, Muller and Satterthwaite (1979) dealt with the case of public goods. The excellent overview of this topic by Le Breton and Weymark (2004) is highly recommended. Geslin, Salles and Ziad (2003) consider the public good case when individual and social preferences are fuzzy. They test the robustness of the results of Barrett, Pattanaik and Salles (1986) when the set of alternatives is the positive orthant of a Euclidean space (that is a pure public good economy where the social and individual preferences are defined over the same set), and when the individuals’ fuzzy strict preferences satisfy some monotonicity properties.

` Geslin, Salles and Ziad (GSZ) consider the case where X = R+, the positive orthant of `-dimensional Euclidean space. They introduce two monotonicity properties on individual BPS-fuzzy strict preferences. Although the ﬁrst is probably speciﬁc to their paper, the second is an adaptation of the monotonicity of preferences of microeconomics texts.

We will use the standard notation regarding inequalities between vectors in R`, i.e., given x = (x1, ..., x`) and y = (y1, ..., y`), x ≥ y if xh ≥ yh for h = 1, ...`; x > y if x ≥ y and x 6= y; and x y if xh > yh for h = 1, ..., `. We will consider BPS-fuzzy strict preferences (see Deﬁnition 2).

Deﬁnition 30. A BPS-fuzzy strict preference p satisﬁes F − monotonicity if for any 0 ` y, x, x ∈ R+, (1) if x ≤ y, p(x, y) = 0, and (2) otherwise, x > x0 ⇒ p(x, y) > p(x0, y) if p(x0, y) 6= 1 and p(x, y) = 1 if p(x0, y) = 1. 5

This deﬁnition is intuitively appealing. It means that the degree to which x is preferred to y is greater (when it is possible) than the degree to which x0 is preferred to y, when x is greater (in the vector sense) than x0.

Let M1 be the set of F -monotonic BPS-fuzzy strict preferences.

5This corrects the deﬁnition in GSZ which is insuﬃcient to prove Theorem 19. 14 RICHARD BARRETT AND MAURICE SALLES

Deﬁnition 31. A F -monotonic BPS-social welfare function is a fuzzy aggregation n function f : M1 → ABPS.

` Theorem 19. Let X = R+. The fuzzy aggregation function f deﬁned by, ` n for all x, y ∈ R+ and all (p1, ..., pn) ∈ M1 , pS(x, y) = (1/n)Σipi(x, y) is a F -monotonic BPS-fuzzy social welfare function satisfying FI and FPC.

Of course, as previously, one can remark that this fuzzy social welfare function satisfies other properties (in particular, properties of symmetry). In BPS (1986), it is indicated that the mean rule is not a BPS-fuzzy social welfare function because (iii) of Definition 3 is not satisfied by the social preference. The preceding result indicates that F -monotonicity is a sufficient condition to obtain (iii). GSZ introduce a second monotonicity property which they call E-monotonicity.

Deﬁnition 32. A BPS-fuzzy strict preference p satisﬁes E-monotonicity if for all ` distinct x, y ∈ R+, x > y ⇒ p(x, y) = 1 and p(y, x) = 0.

This is, of course, similar to the standard property of microeconomics. Let M2 be the set of E-monotonic BPS-fuzzy strict preferences.

Deﬁnition 33. An E-monotonic BPS-social welfare function is a fuzzy aggregation n function f : M2 → ABPS.

GSZ show that Theorems 1 and 2 are essentially preserved when individual fuzzy preferences are E-monotonic BPS-fuzzy strict preferences.

` 0n n Theorem 20. Let X = R+ and f : M2 ⊆ M2 → ABPS be an E-monotonic BPS-fuzzy social welfare function satisfying FI and FPC and having a non-narrow domain for all ` distinct x, y, z ∈ R+ such that there is no >-relation between any two of them. Then there exists a unique coalition C such that ` 0n for all distinct x, y ∈ R+ and all (p1, ..., pn) ∈ M2 , if pi(x, y) = 1 and pi(y, x) = 0 for every i ∈ C, then pS(x, y) > 0, where pS = f(p1, ..., pn); and 0n for all distinct x, y ∈ X and all (p1, ..., pn) ∈ M2 , if for some j ∈ C, pj(x, y) = 1 and pj(y, x) = 0, then pS(y, x) = 0, where pS = f(p1, ..., pn).

The following theorem is a simple corollary (in the same way as Theorem 2 is a corollary of Theorem 1).

` 0n n Theorem 21. Let X = R+, f : M2 ⊆ M2 → ABPS be an E-monotonic BPS- fuzzy social welfare function satisfying FI and FPC and having a non-narrow domain for ` all distinct x, y, z ∈ R+ such that there is no >-relation between any two of them, and 0n pS ∈ f(M2 ) be BPS-complete. Then #C = 1: there exists an individual i ∈ N such that ` 0n for all distinct x, y ∈ R+ and all (p1, ..., pn) ∈ M2 , if pi(x, y) = 1 and pi(y, x) = 0, then pS(x, y) > 0 and pS(y, x) = 0, where pS = f(p1, ..., p,).

These two theorems can be stated and proved in an essentially similar manner for another monotonicity assumption: F ?-monotonicity. p is said to be F ?-monotonic if for all ` distinct x, y ∈ R+, x > y ⇒ p(x, y) > 0 and p(y, x) = 0.

GSZ consider another subclass of E-monotonic BPS-fuzzy strict preferences, which appears as very restrictive and rather arbitrary. Their purpose, here, is to show that they obtain similar results even in this very restrictive case, which exempliﬁes the robustness of their quasi-negative results. SOCIAL CHOICE WITH FUZZY PREFERENCES 15

` Deﬁnition 34. Let a ∈ R+, a 0, and α ∈]0, 1[. The strictly positive vector a is said aα ` ` to α-parametrize a fuzzy binary relation p over R+ if for all x, y ∈ R+,

  1 if ax > ay (4.1) paα (x, y) = α if ax = ay and x 6= y  0 otherwise, where ax and ay are dot products.

` Let P be the set of fuzzy binary relations over R+ that are α-parametrized by some strictly positive vector for some α ∈]0, 1[.

Deﬁnition 35. A P − BPS-social welfare function is a fuzzy aggregation function n f : P → ABPS.

Theorem 22. If p ∈ P, then p ∈ M2.

0 We now show again that Theorems 1 and 2 are essentially preserved when ABPS = P.

` n Theorem 23. Let X = R+ and f : P → ABPS be a P − BPS-fuzzy social welfare function satisfying FI and FPC. Then there exists a unique coalition C such that ` n for all distinct x, y ∈ R+ and all (p1, ..., pn) ∈ P , if pi(x, y) = 1 and pi(y, x) = 0 for every i ∈ C, then pS(x, y) > 0, where pS = f(p1, ..., pn), and n for all distinct x, y ∈ X and all (p1, ..., pn) ∈ P , if for some j ∈ C, pj(x, y) = 1 and pj(y, x) = 0, then pS(y, x) = 0, where pS = f(p1, ..., pn).

The following theorem is a simple corollary (in the same way as Theorem 2 is a corollary of Theorem 1).

` n Theorem 24. Let X = R+, f : P → ABPS be a P-BPS-fuzzy social welfare function n satisfying FI and FPC and pS ∈ f(P ) be BPS-complete. Then #C = 1: ` there exists an individual i ∈ N such that for all distinct x, y ∈ R+ and all (p1, ..., pn) ∈ n P , if pi(x, y) = 1 and pi(y, x) = 0, then pS(x, y) > 0 and pS(y, x) = 0, where pS = f(p1, ...pn).

5. concluding remarks It is clear from this chapter (which we hope is as complete as possible) that there are many routes that have not yet been explored within fuzzy set theory. The most striking feature of this approach is its flexibility. Exploring various assumptions each with intuitive appeal, one can obtain divergent results. This is true when we consider different ways to decompose fuzzy weak preferences. It is also true, in economic environments, when we consider different monotonicity conditions.

Finally, let us mention that fuzzy sets are only one way to deal with imprecision and vagueness. Other approaches are possible, even though largely unexplored in economic theory. This is the case with rough sets theory which provides a nice way to describe similarities (indiﬀerences). It is also the case with supervaluation theory , particularly in vogue among philosophers, as mentioned in the introduction, and with the theory of interval orders (Fisburn (1985)) and other aspects of measurement theory.

First, let us consider rough sets. Given a set X and a partition of X (geometrically, one can imagine some kind of grid), any subset S of X will possibly contain elements of the 16 RICHARD BARRETT AND MAURICE SALLES partition (that is, entire subsets belonging to the partition). The union of these elements will be considered as an inner approximation of S (it will be included in S). Also, if we consider elements of the partition that have a non-empty intersection with S, one can take the union of these elements. This union will contain S and will be the outer approximation of S. This construction due to Pawlak (1982) (see also Polkowski (2002)) has been used by Bavetta and del Seta (2001) to describe problems in the freedom of choice literature, in particular to deal with the diﬃculty raised by indistinguishable alternatives.

In supervaluation theory (a theory pertaining to philosophical logic), a proposition con- taining a vague term is true if it is true in all sharpenings of the term. A ‘sharpening’ is an ‘admissible’ way in which a vague term can be made precise. Consider, for instance, the term ‘old.’ ‘Old’ can be interpreted as ‘being over 55 years of age.’ Alternatively, it can be interpreted as ‘being over 60 years of age,’ and so on. None of these ‘sharpenings’ is the actual meaning of ‘old,’ which is vague, but they are ways ‘old’ might be sharpened. On the other hand, ‘being over 20 years of age’ cannot be a ‘sharpening’ of ‘old’ since it is clearly not an ‘admissible’ way of making ‘old’ precise. So the notion of ‘being admissible’ is crucial. For supervaluation theory and the (philosophical) study of vagueness Williamson (1994), Keefe (2000) and Piggins (1999) are recommended. We have drawn our presentation from Piggins (1999) which also included applications to welfare economics.

Finally, in measurement theory (Fisburn (1970), (1988), Krantz, Luce, Suppes and Tversky (1971)), a binary relation ST on X ×X, where X is the set of alternatives, is interpreted as a comparison of the strengths of preference between ordered pairs, (x, y) ST (z, w) meaning that the strength of preference of x over y exceeds the strength of preference of z over w. An underlying preference over X is defined by x y if (x, y) ST (y, y). Given a set of axioms, a utility representation for comparable differences, i.e., a real-valued function u such that (x, y) ST (z, w) ⇔ u(x) − u(y) > u(z) − u(w) can be derived. This, of course, is reminiscent of the von Neumann-Morgenstern representation of a complete preorder over lotteries, but despite this the formalization has been, to the best of our knowledge, largely ignored by fuzzy set theorists. (A problem is that it is also rather difficult to interpret the notion of strength of preference of y over y. We would need for this to have a notion of (unique) minimum for the strength of preference relation.)

References Arrow, K.J. (1963) Social choice and individual values, Second Edition. New York: Wiley.

Atanassov, K.T. (1999) Intuitionistic fuzzy sets: theory and applications. Heidelberg: Physica- Verlag.

Banerjee, A. (1994) ‘Fuzzy preferences and Arrow-type problems in social choice’, Social Choice and Welfare, 11: 121-130.

Barrett, C.R. and P.K. Pattanaik (1990) ‘Aggregation of fuzzy preferences’, in: J. Kacprzyk and M. Fedrizzi, eds., Multiperson decision making using fuzzy sets and possibility theory. Dordrecht: Kluwer.

Barrett, C.R., P.K. Pattanaik and M. Salles (1986) ‘On the structure of fuzzy social welfare functions’, Fuzzy Sets and Systems, 19: 1-10.

Barrett, C.R., P.K. Pattanaik and M. Salles (1990) ‘On choosing rationally when preferences are fuzzy’, Fuzzy Sets and Systems, 34: 197-212.

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department of economics, university of birmingham, Birmingham B15 2TT, united kingdom E-mail address: [email protected] SOCIAL CHOICE WITH FUZZY PREFERENCES 19

gemma, umr-cnrs 6154, mrsh, universite´ de caen, 14032 caen cedex, france and cpnss, london school of economics, houghton street, london wc2a 2ab, united kingdom E-mail address: [email protected]